So Brilliant.org says this: Since the difference between Pn and Pn+1 is just that last term [of Pn+1], the error of Pn can be no larger than that term. I get how this works for alternating series, because the polynomial "crosses" over the function value every time you add a term (if function value is 5, it might go 1, then 7, 4, then 5.5), so your the difference from that term is greater than the difference of your polynomial from the root function. But what about positive series like e^x? With that, if your actual value is 5, you get 1, then 3, then 4, then 4.5, etc. So the difference between a four term and a five term polynomial is less than the difference between the four term and the actual function. So clearly the error is greater than that. Am I missing some way that this is somehow correct? Or are they wrong in that particular case? and if so, is there a way to actually find the error bound?
I have to teach a lesson on taylor series error bound in a few days and not understanding this part is really bugging me.